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Edit. As t3suji points out, the OP wants an example where $\mathcal{F}_n(U)$ equals $\varinjlim\left(\mathcal{F}_n(U)\right)$, which my example does not satisfy. I suspect that one can make a new example of a sequence of short exact sequences $0\to \mathcal{F}_n \mathcal{G}_n\to \mathcal{Q}_n \to 0$, where the sheaves $\mathcal{F}_n$ do satisfy the OP's constraint, yet the sheaves $\mathcal{Q}_n$ do not. In that case, the long exact sequence of cohomology would imply that the groups $H^1(X,\mathcal{F}_n)$ are not compatible with colimits. I will try to find an example; until then, I am leaving my original example.

Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every nonnegative integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the closed interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$. For each $n$, the quotient of $\mathcal{O}_X$ by the subsheaf $\mathcal{F}_n$ is the sheaf $(i_n)_*\mathcal{O}_X|_{Z_n}$ of germs of holomorphic functions on neighborhoods of $

Since stalks commute with colimits, for every $p\in U$, for every $m\geq 0$, $$(\mathcal{F}_m)_p = \varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p.$$ Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.

Edit. As t3suji points out, the OP wants an example where $\mathcal{F}_n(U)$ equals $\varinjlim\left(\mathcal{F}_n(U)\right)$, which my example does not satisfy. I suspect that one can make a new example of a sequence of short exact sequences $0\to \mathcal{F}_n\to \mathcal{G}_n\to \mathcal{Q}_n \to 0$, where the sheaves $\mathcal{F}_n$ do satisfy the OP's constraint, yet the sheaves $\mathcal{Q}_n$ do not. In that case, the long exact sequence of cohomology would imply that the groups $H^1(X,\mathcal{F}_n)$ are not compatible with colimits. I will try to find an example; until then, I am leaving my original example. Actually, having failed to construct an example, I tend to agree with Mikhail Bondarko. There is the issue that when you form Čech complexes, you typically need to take infinite products, and these do not commute with colimits. But at least for sheaves that have a finite acyclic cover, it does seem that the Čech complex proves that all cohomology commutes with colimits, so long as the colimit presheaf is a sheaf. As before, I am leaving my original example, which proves that the colimit presheaf is typically not a sheaf.

Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every nonnegative integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the closed interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$. For each $n$, the quotient of $\mathcal{O}_X$ by the subsheaf $\mathcal{F}_n$ is the sheaf $(i_n)_*\mathcal{O}_X|_{Z_n}$ of germs of holomorphic functions on neighborhoods of $

Since stalks commute with colimits, for every $p\in U$, for every $m\geq 0$, $$(\mathcal{F}_m)_p = \varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p.$$ Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.

Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every nonnegative integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the closed interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$.

Since stalks commute with colimits, for every $p\in U$, for every $m\geq 0$, $$(\mathcal{F}_m)_p = \varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p.$$ Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.

Edit. As t3suji points out, the OP wants an example where $\mathcal{F}_n(U)$ equals $\varinjlim\left(\mathcal{F}_n(U)\right)$, which my example does not satisfy. I suspect that one can make a new example of a sequence of short exact sequences $0\to \mathcal{F}_n \mathcal{G}_n\to \mathcal{Q}_n \to 0$, where the sheaves $\mathcal{F}_n$ do satisfy the OP's constraint, yet the sheaves $\mathcal{Q}_n$ do not. In that case, the long exact sequence of cohomology would imply that the groups $H^1(X,\mathcal{F}_n)$ are not compatible with colimits. I will try to find an example; until then, I am leaving my original example.

Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every nonnegative integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the closed interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$. For each $n$, the quotient of $\mathcal{O}_X$ by the subsheaf $\mathcal{F}_n$ is the sheaf $(i_n)_*\mathcal{O}_X|_{Z_n}$ of germs of holomorphic functions on neighborhoods of $

Since stalks commute with colimits, for every $p\in U$, for every $m\geq 0$, $$(\mathcal{F}_m)_p = \varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p.$$ Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.

Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$.

Since stalks commute with colimits, for every $p\in U$, $\varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p$. Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.

Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every nonnegative integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the closed interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$.

Since stalks commute with colimits, for every $p\in U$, for every $m\geq 0$, $$(\mathcal{F}_m)_p = \varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p.$$ Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.